How can I find a function, $$ f: \mathbb{R} \to \mathbb{R} $$ which satisfies the following equation:
$$\cos\left(t^2\right) = \int_{-\infty}^{\infty} e^{itx}f(x)\,dx$$
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Perhaps you can't. Even if it exists, that does not mean that it can be expressed in terms of known functions, or analytic series, or even as a definite integral of parameter $t~($like the beta and $\Gamma$ functions, for instance$)$. – Lucian Aug 31 '15 at 13:22
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@Lucian - Actually, I am simply trying to show that all solutions to this equation (if any exist at all) are not probability distributions. – Kerry Aug 31 '15 at 15:39